Radiation-redistributive devices

ABSTRACT

Devices for redistributing incident radiation in a prescribed manner. Such devices comprise a multitude of contiguous optical microelements, each of such microelements being contoured and oriented to redistribute incident radiation, emanating from an intended irradiating source, only throughout an angular field just large enough to encompass a predefined region wherein the redistributed radiation has particular utility. Moreover, the contour and orientation of each microelement is such as to produce substantially uniform radiance throughout such predefined region of utility, and to redirect extraneous or undesirable radiation incident thereon away from said predefined region. The radiation-redistributive devices of the invention are particularly useful as front or rear projection screens, lighting reflectors or refractors, illumination aids for photographic prints, traffic signs, advertisements, etc, etc.

United States Paten Depalma et al.

[ Aug. 28, 1973 1 RADIATION-REDISTRIBUTIVE DEVICES [75] Inventors: JamesJ. Depalma; Harold F.

Langworthy, both of Rochester, NY.

[73] Assignee: Eastman Kodak Company,

Rochester, N.Y.

[22]- Filed: Dec. 13, 1971 [21] Appl. No.: 207,082

52 us. Cl. 350/127, 350/126 [51] Int. Cl. G031) 21/60 [58] Field ofSearch ..350/l 17-129 [56] References Cited UNITED STATES PATENTS2,804,801 9/1957 Mihalakis 350/129 2,207,835 7/1940 Sukumlyn.... 350/1171,970,358 8/1934 Bullet a1. 350/127 2,510,344 6/1950 Law 350/1282,480,031 8/1949 Kellogg 350/127 Primary Examiner-Samuel S. MatthewsAssistant Examiner-Richard M. Sheer Attorney-Robert W. Hampton et al.

[57] ABSTRACT Devices for redistributing incident radiation in aprescribed manner. Such devices comprise a multitude of contiguousoptical microelements, each of such microelements being contoured andoriented to redistribute incident radiation, emanating from an intendedirradiating source, only throughout an angular field just large enoughto encompass a predefined region wherein the redistributed radiation hasparticular utility. Moreover, the contour and orientation of eachmicroelement is such as to produce substantially uniform radiancethroughout such predefined region of utility, and to redirect extraneousor undesirable radiation incident thereon away from said predefinedregion. The radiation-redistributive devices of the invention areparticularly useful as from or rear projection screens, lightingreflectors or refractors, illumination aids for photographic prints,traffic signs, advertisements, etc, etc.

18 Claims, 21 Drawing Figures Patented Aug 28,1973 3,754,813

'9 Sheets-Sheet l JAMES J. DePALMA HAROLD l-fLAAlawmr/lr INVENTORS wwz/ATTORNEY 9 Sheets-Sheet 2 RAD/A/VC E m) F/Gf 6 JAMES J. DBPALMA 5 HAROLDFLA/VGWORTHY INVENTORS Arrows) Patented Aug. 28, 1973- 3,754,813

1) Sheets-Sheet a JAMES J DePALMA HAROLD F LA/VGWORTH) INVENTORSATTORNEY Patented Aug; 28, 1 973 v I muss J DePAL/HA HAROLD FLA/VGWORTH) INVENTORS mud/g ATTORNEY Patented Aug. 28, 19 73 3,754,813

9 Sheets-Sheet 5 205 FIG /8 JAMES J. DePALMA HAROL D F LA/VGWOR TH YINVENTORS D/RE' C 7' ION 0F WOR/(P/ECE BY TRAVEL &

Patented AugQZS, 1973 9 SheetsS heet 9 .mb i km QR 1 N VENTORS QQN QQKQQIMRWQ MS wk MEG ATTORNEY RADIATION-REDISTRIBUTIVE DEVICESCROSS-REFERENCE TO RELATED APPLICATIONS References made to the commonlyassigned copending applicationSer. No. 207,334 entitled Method andApparatus for Fabricating Radiation-Redistributive Devices, filedconcurrently herewith in the names .of Robert N. Wolfe, et al.

BACKGROUND OF THE INVENTION In ,general, the present invention relatesto improvements in projection screens and other radiation-;redistributive devices, such as .lighting reflectors and refractors,illumination aids for photographic prints,

:traffic signs, advertisements, etc.

:Heretofore, a wide variety of radiation-redistributive devices has beenproposed to achieve such features as a definitely controllable fieldthrough which incident radiation is redistributed, uniform radiancethroughout such field, high efficiency due ;to a definite separation .ofthe field of redistribution from the environmental field and due tominimum absorption losses at the redistributing surface of such devices,and a favorable rejection of radiation impinging on the device fromsources other than those intended for irradiating the device.

In attempting to provide radiation-redistributive devices having one ormore of the features mentioned above, twoapproaches have been taken. Oneapproach is purely empirical in nature and involves the evaluation ofcommercially available, inherently reflective,

refractive, or diffuse materials to determine the utility thereof for aparticular application. Exemplary of the projection screens developedthrough such an empirical approach are the volume and surfacediffuser-type rear projection screens, and the aluminum foil frontprojection screens disclosed in the commonly assigned U. 'S. Pat.- No.3,408,132, high reflectance projection screens commercially availableunder the trademark Kodak Ektalite Projection Screen.

The second approach toward the provision of improvedradiation-redistributive devices is analytical in nature, involving thederivation of mathematical expressions to define the contour which eachelemental area of the redistributing surface must possess in order toachieve a desired redistribution of incident radiation, and thefabrication of an optical surfaces in accordance with such expressions.Lenticular projection screens and general lighting refractors areexemplary of such an analytical approach.

Notwithstanding the approach taken, radiationredistributive devicesheretofore proposed have not been totally satisfactory in all respects.Usually, certain desirable features are severely compromised to achieveother features which are deemed more desirable for a particularapplication. For instance, in the case of projection screens, severalscreens have been proposed having reflecting or refracting surfaceswhich, at least in theory, are capable of redistributing incident-imagelight in such a manner that the luminance of every elemental area on thescreen surface is substantially constant throughout a predefined angularfield of observation. Such screens, however, often suffer thedisadvantages of being inefficient or wasteful of available image lightand of being difficult, if not, for all practical purposes, totallyimpractical to manufacture in large quantities. See, for instance, thescreens disclosed in U. S.

Pat. No. 3,257,900 and U. S. Pat. No.. 2,870,673. On the other hand,projection screens having highly efficient and readily manufacturablesurfaces are often incapable of distributing incident image lightuniformly and in a controlled manner, such surfaces commonly exhibitinghot spots or regions of non-uniform luminance.

SUMMARY OF THE INVENTION Accordingly, one of the objects of the presentinvention is to provide a radiation-redistributive device which not onlymakes maximum utilization of radiation emanating from an intendedirradiating source by redistributing substantially all of such radiationonly throughout a region wherein the redistributed radiation .hasutility, but also redistributes incident radiation in such a manner thatsubstantially uniform radiance is produced throughout such region ofutility.

Another object of the invention is to provide aradiation-.redistributive device of the above type which, upon beingirradiated by extraneous or nondesirable radiation emanating from asource other than that intended for irradiating the device,redistributes or redirects such nondesirable radiation away from saidregionof utility, thereby maximizing the signal-to-noise ratio for anyenvironmental lighting condition.

A further object of this invention is to provide raida--tion-redistributive devices of the above type which are readilymanufacturable in large quantities.

.In accordance with the present invention, the above objects areachieved by the provision of a radiationredistributive device whichcomprises a plurality of contiguous optical microelements, each beingcontoured substantially in accordance with a mathematical expressionrequiring that all radiation incident thereon be redistributed therefromsuch as to produce uniform radiance throughout a predefined angularfield and substantially zero radiance outside such field. Preferably,the radiation-redistributing surface of such device is planar and eachof said microelements comprising said surface is uniquely contoured andarranged, depending upon its respective position on the surface, toredistribute incident radiation only throughout a solid angle just largeenough -to encompass a region which is intended to receive radiationfrom said device. Alternately, the redistributing surface of the deviceis substantially spherically or cylindrically curved and all of themicroelements contoured substantially alike, the screen curvatureassisting in redistributing incidentradiation into the designed region.

In addition to the objects of the invention set forth herein above,other objects and advantages of the invention will become apparent tothose skilled in the art from the ensuing description, reference beingmade to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIGS. 1 and 2 are, respectively,diagrammatical representations of rear and front projection systems,respectively, illustrating the radiation-redistributing properties ofrear and front projection screens embody ing the invention;

FIG. 3 shows the ideal radiance curve for a radiationredistributivedevice;

FIGS. 4-6 illustrate a reflective radiationredistributive device,structured in accordance with a and sections taken along lines 5-5 and6-6, respecredistribution FIG. 8 is a constructional diagram of themechanism used to drive the cutting stylus of the sound recording headillustrated in FIG. 7; 7

FIG. 9 is a side elevational view of the cutting stylus and supporttherefor;

FIG. 10 is a front elevation of the cutting stylus illustrating thecutting profile thereof;

FIG. 11 is a perspective view of apparatus for translating a master fromwhich the radiation-redistributive devices can be subsequentlyreplicated relative to the cutting apparatus illustrated in FIG. 7;

FIG. 12 illustrates the manner in which the waveform applied to thecutting stylus differs from the sytlus motion produced thereby;

FIG. 13 is an electrical schematic of circuitry for driving the cuttingstylus to produce one side of the radiation-redistributive surfaceillustrated in FIGS.

' FIG. 14 illustrates the additional logic circuitry required to modifythe circuit of FIG. 13 so as to produce the entire surface illustratedin FIGS. 4-6;

FIG. '15 illustrates the manner in which the positivegoing ramp of asawtooth waveform is shaped to a desired stylus-driving waveform;

FIG. 16 illustrates the input signal to the cutting stylus when in acuttingposition displaced from center of the master;

FIGS. 17-19 illustrate a radiation-redistributive device structured inaccordance with another preferred embodiment of the invention, in planview and in section taken along lines l8--18 and 19-19, respectively;and

FIGS. 18a and 19a illustrate alternative cross sections for the devicesillustrated in FIG. 17.

FIG. 20 is an electrical schematic of circuitry adapted to drive thecutting stylus in a manner such as to produce a master from whichradiationredistributive devices, such as that illustrated in FIGS.

1749, can be fabricated.

DETAILED DESCRIPTION OF PREFERRED 1 EMBODIMENTS As'indicated above, theradiation-redistributive devices of the invention have utility in anysituation wherein it is desirable or necessary to precisely control theredistribution or redirection of radiant energy from a surface which isirradiated by a source which occupies a predictable position relative tosuch surface.

- Such devices have been found to have particular utility as projectionscreens, both of the front and rear projection variety, being capable ofpresenting to a precisely definable audience area or field ofobservation a projected image of substantially uniform luminance, re-

, gardless of the viewing position within such audience area.Accordingly, projection screens represent a preferred embodiment of theinvention and the invention will be described hereinafter withparticular reference thereto; however, it should be borne in mind thatpro jection screens are merely exemplary of the uses for the invention,and all such uses, including those previously mentioned, are consideredwithin the spirit and scope of the invention.

The radiation-redistributing surface of the invention devices wasderived mathematically based upon the theorems of geometrical optics andupon the following postulates which are believed to define aradiationdevice having ideal radiationredistributive properties:

I. Every elemental area on the redistributing surface of the deviceshall redistribute all radiation incident thereon from an intendedirradiating source throughout a solid angle just large enough toencompass a predefined field wherein the redistributed radiation hasutility. (A radiation-redistributing surface satisfying this postulateis one of maximum efficiency, utilizing all available radiation).

II. Every elemental area on the radiationredistributing surface of thedevice shall redistribute incident radiation in such a manner that theradiance .of every such area will be constant, no matter where measuredwithin the solid angle through which radiation is redistributed. (Withrespect to projection screens, a surface satisfying this postulate isone which will present an image of uniform brightness everywhere withinthe intended audience area, and no image whatsoever outside suchaudience area).

In FIGS. 1 and 2, projection systems comprising projectors P and rearand front projection screens S and S, respectively, are shown in twodimensions. Image light or flux emanating from the projector is focusedupon the surfaces l0-of the projection screens, rear projection screen Sbeing fabricated from a transparent material. As shown, eachsurface-comprises a plurality of contoured optical microelements' 11,shown, forpurposes of illustration, greatly magnified and of concaveshape. Actually,'each microelement is preferrably of a size so as to beunresolvable by the closest intended viewer and can be either concave orconvex in shape. The contour of eachmicroelement is such that image fluxincident thereon at an angle 0, measured from a line connecting theprojector lens 15 to the screen center C, is redistributed throughout aviewing angle H which 'is just'large enough 'to encompass the predefinedaudience volume V. Viewing angle H consists of left and right viewingangles A and-B, respectively, each being measured from a normal N to thescreen surface. All angles are considered positive when measured in acounterclockwise direction from a nonnal to the screen surface;Because'the size of the. microelement is quite small relative to thedistance separating the microelement from the projector, all light raysstrikinga particular microelement are assumed to be parallel.

-'It can be shown mathematically that to satisfy Postuwhere n is therefractive index of the microelement (n being 1 when the microelement isreflective); u and w are the microelement coordinates, w being measuredin-a direction parallel to the path of the rays of radiation incident onthe microelement and u being measured in the plane of the cross section,perpendicular to w; and w has a value within the following limits:

cos (A w l, whenflwm) is positive and the microelement refractive; cos(B 0') 5 w s 1, when f(w;n) is negative and the microelement refractive;cos (B 0) s w s l, whenflwm) is positive and the micrpelementreflective; cos (A 0) 5 w s l, whenf(w;n) is negative and themicroelement reflective; where 0 is the projection in the u w plane ofthe angles formed by a line connecting the microelement and projector,and the normal .to the screen surface; and n sin 0/sin 0.

Similarly, it can be shown that every concave optical microelementcomprising the projection screen surface must have transverse andlongitudinal cross sections defined by at least a portion of the curvewherein w has a value within the following limits:

1 s w s -cos (B 0'), when g(w;n) is positive and the microelement isrefractive; --l s w s cos(A 0' when g(w;n) is negative and .themicroelement is refractive; i s w s cos (A 0), when g(w;n) is positiveand the microelement is reflective; and

6 ,where concave and is. defined by Equation (2) above. The depthprofile or longitudinal cross section of each groove, as illustrated inFIG. 6, alternately varies or undulates in shape from convex to concave,the convex and concave portions being defined by Equations (1) and (2),respectively. Thus, for each full wavelength of depth undulation, twomicroelements are formed, one concave and one convex. The longitudinalboundaries of each microelement are the lines 18 along which the l s w scos'(B 0), when g(w;n) is negative and the microelement'is reflectiveEquations (1) and (2) above define the contour required for amicroelement to provide an ideal goniphotometric response (i.e.,constant radiance, between horizontal audience angles A and B, andvertical audience angles A and B. (See FIG. 3). As is apparent fromthese equations, screen performance is independent of the size and sense(concave or convex) of each microelement relative to adjacent ones.Thus, the microelements could be of random sizes and randomly sensedover the entire screen surface so long as the contours defined byEquations (1) and/or (2) are substantially met, such contours beingdependent only on the angle at which incident light impinges themicroelement surface, the horizontal and vertical audience an-v glesthrough which such light must be distributed to encompass the intendedaudience area, and, in the case of rear projection screens, therefractive index of the material from which the microelements areformed. l lowever, to facilitate the manufacture of such screen, it ispreferred that microelements be substantially the same in size and bearranged in contiguous linear rows, each microelement having atransverse cross section which is everywhere curved in the same sense,prefer- -rably concave, and a longitudinal cross section which is shapedin an opposite sense from that of adjacent microelements in the samerow. Thus,the preferred longitudinal cross section of each row ofmicroelements is one which undulates from concave to convex, etc. Such asurface is illustrated in FIGS. 4 through 6.

I shown in FIGS. 4 through 6, a screen structured in accordance with thepresent invention comprises a surface which defines a plurality ofcontiguous linear edges of each groove intersect with those of adjacentgrooves, cusp lines 17 are formed. The transverse cross section of eachgroove, as depicted in FIG. 5, is everysense of the depth profilechanges from concave to convex, or vice versa. The lateral boundaries ofeach microelement are, of course, provided by cusp lines 17. As bestshown in the sectional views of FIGS. 5 and 6, the microelements aregradually tilted as their re- ,spective displacement from screen centerC increases. The degree of such tilt is determined by the size and shapeof th viewing area and, hence, the values of viewing angle A and B. Thesense of such tilt (i.e., toward or away from .the screen center)depends on whether the screen is reflective or refractive, reflectivemicroelements being tilted toward the screen center, as shown in FIGS.5a, 50, 6a, and 6c, and refractive microelements being tilted away fromthe screen center. Generally, the microelements are oriented withrespect to each other such that the lines, formed by intersecting planeswhich bisect the horizontal and vertical angles through which eachelement distributes image light to the the audience, substantiallyintersect at a common point in the audience area.

To fabricate the radiation-redistributive devices of the invention, ithas been found that various equipment and techniques conventionallyemployed in the sound recording industry can be used directly or in amodified form. In FIG. 7, apparatus used for cutting theradiation-redistributive microelements is illustrated in a sideelevatiorial view, being shown in a cutting position relative to a blankworkpiece 20 wherein microelements are to be formed. While themicroelements could be out directly in any readily workable materialwhich itself could be used as the radiation-redistributive device, thepreferred method of manufacture comprises the fabrication of a master insome workable material,

such as acetate or wax, from which a negative matrix or press tool ofcorrect contour can be subsequently made. The negative matrix can thenbe used to produce a multitude of positive radiation-redistributivedevices by such well-known economical duplicating processes as stampingor embossing. Preferred methods of replicating radiation-redistributivedevices frommasters are described subsequently herein.

As shown in FIG. 7, the cutting apparatus comprises a conventionalstereo-sound recording head 30 which includes a cutting stylus S. Whilea monaural sound recording head could be used, a stereo head ispreferred due to the high quality of conventional stereo heads and theauxiliary equipment available for such stereo heads. As in all soundrecordings heads, the cutting position of the stylus is determined bythe waveform of an electrical signal applied to the recording head, suchas through input cables 31. The recording head is mounted on a millingmachine tool holder 32 by a cylindrical fitting 33. Means are providedfor controlling the vertical position of fitting 33 in the tool holder32 so as to provide a coarse, vertical adjustment of the recording head30 above the workpiece. The workpiece may comprise, for instance, analuminum plate 36 having an acetate coating 37, the thickness of whichis sufficient to receive the contours of the microelements being cut.Recording head 30 comprises a cutting assembly 40 having a horizontallyextending support arm 41 which is slidably mounted on precision waysdisposed in a saddle 42. By this arrangement, the horizontal position ofcutting assembly 40 can be varied. Set screws 43a and 43b serve to lockarm 41 .in a desired horizontal position. Saddle 42 is pivotally mountedabout pin 44 disposed on recording head 30 so that the cutting stylus S,which forms a part of cutting assembly 40, can be moved into engagementwith coating 37. The rotational move-ment of a cam 46 serves to raiseand lower the stylus relative to the upper surface of coating 37 bycontacting an arm 47 which is rigidly coupled with saddle 42. Thedownward force applied to the cutting assembly is controlled by screw 48which serves to adjust the tension in spring 49. The precise depth ofcut is controlled by adjustment screw 50 which varies the verticaldistance of the stylus tip from a small glass ball follower 51 whichrides on the uncut surface of coating 37 a short, horizontal distanceaway from the stylus.

A sound recording head which has been found particularly well adaptedfor cutting projection screen masters is the Westrex Corporation, Model3D StereoDisc.

As illustrated in FIG. 8 wherein a simplified constructional diagram ofthe mechanism which controls stylus movement is shown, each recordingchannel of the stereo recording head contains a magnetic coil formassembly 60, each of which contains a driving coil 62 located inseparate pole pieces 64 and 65which are at-- tached to a single magnet66.

The coil assemblies are attached to the stylus holder through links 68which are stiff longitudinally, but flexible laterally. These links arebraced in the center to prevent excessive lateral compliance. Thisstructure results in a stiff, forward driving system with a highcompliance in the lateral direction. v i

The supporting member for the stylus is shown in FIG. 9. The use of acantilever spring 70 permits the stylus to present a uniform impedanceto complex motions in any direction in the vertical plane perpendicularto the drawing. i

The cutting tip 72 of stylus S is preferably fabricated from sapphire ordiamond, and is heated by heating coil 73 to a temperature such as tosoften the acetate surface of the workpiece. If the surface described bythe above equations were to be produced exactly, it would be necessaryto use a stylus having a different cutting profile for each groove cut.However, it has been found that when the intended audience area can beencompassed by audience angles of less than approximately :40 degreesmeasured from the normal, the ideal screen surface can be satisfactorilyapproximated by using a single cutting stylus having a cutting profiledefined by Equation (1) at 0 0, and by tilting the stylus axis duringthe cutting operation so as always to be parallel to the plane whichbisects the audience angle through which the groove being cut mustdistribute image flux to encompass the intended viewing area. Therequired curve is illustrated by the cutting profile of the styluscutting tip 72 in FIG. 10, such curve being somewhat flattened relativeto a half sinusoid.

In fabricating projection screen masters by use of the apparatusdescribed above, the workpiece is moved relative to the heated cuttingstylus in a series of equally spaced, parallel traverses. At the sametime, the cutting I position of the stylus is electronically variedrelative to the workpiece surface to produce the desired longitudinalcross section or depth profile. Apparatus for moving the workpiecerelative to the stylus is depicted in FIG. 11. As shown, such apparatuscomprises a table for supporting the workpiece during the cuttingoperation. Table 80 is preferably fabricated from a nonmagnetic metal,such as aluminum, so as not to interfere with the magnetic cuttingassembly 40. In the upper surface of table 80, a circular groove isprovided. At the base of groove 85 is an opening (not shown) whichcommunicates with a nozzle 86 located on the edge of the table. Attachedto nozzle 86 via hose 87 is a vacuum source (not shown). By thisarrangement, the workpiece is securely fastened to the surface of table80 by a vacuum coupling. Table 80 is mechanically secured to a movableworkbed 88 comprising the x-y table 89 of a milling machine. workbed 88is movable in the x direction and its position is controlled withprecision by a conventional stepping motor 90 which acts through leadscrew 91. workbed 88 itself rides atop a carriage 93, also forming apart of the x-y table of the milling machine. Carriage 93 is movable inthe y direction by a hydraulic pneumatic motor 95 which preciselycontrols the rate at which the table moves via piston rod 96.

To fabricate projection screen masters having trans verse andlongitudinal cross sections such as depicted in FIGS. 5 and 6,respectively, it is necessary to drive the cutting stylus with a signal,the waveform of which varies in accordance with the y-position of thestylus on the screen blank surface. Moreover, as mentioned above, it isalso necessary to vary the angle at which the stylus contacts the screenblank in accordance with the x-position of the stylus on the workpiecesurface.

To maintain the proper orientation between the cutting stylus and theworkpiece during movement of the workpiece in the x direction, themilling machine tool holder is motorized so as to be capable of tiltingthe recording head in the x-z plane in accordance with an electricalinput signal. The x-z position of the stylus is changed after eachgroove is cut so that at all times during the cutn'ng operation theangle t between the longitudinal axis of the stylus and the work surfaceis:

where x is measured from the workpiece center, and p is the projectionin the x-z plane of the distance from the workpiece, center to the lineatwhich the planes which bisect the requisite audience angles of the cutgrooves Fsubstantially intersect. The motorized tool holder of themilling machine is controlled by the output of stepping motor 90.

To move the cutting stylus in a vertical plane and at a rate which, whenthe workpiece is moved at a constant rate relative thereto, results inthe longitudinal cross section or depth profile desired, the same signalmust be applied, out of phase, to both drive coils 62. Moreover, sincethe stylus is not mounted for pure vertical movement, but rather forpivotal movement on the cantilever spring 70, so as to traverse anarcuate path as shown in phantom lines, it is necessary to drive thestylus with a somewhat different waveform than that which corresponds tothe longitudinal cross section desired. Referring to FIG. 12, when awaveform 101 isapplied to the cutting stylus, the resulting groove has adepth profile as shown by the asymmetrical waveform 102. To compensatefor the asymmetry, it is necing asymmetrical waveform 103 which thearcuate styl'us movement converts to the depth profile desired (e.g.,waveform 101). It is intersecting to note that in the sound recordingart, such asymmetry is automatically compensated for during playback bythe pickup stylus, which is also pivotally mounted and moves along anarcuate path similar to that along which the stylus which cut theoriginal master moved. In achieving'a desired profile in projectionscreens, however, such asymmetry must be compensated for by circuitryfor modifying the desired waveform accordingly.

Circuitry for driving the cutting stylus to produce a depth profile suchas illustrated in FIG. is illustrated in FIGS. 13 and 14. To facilitatean understanding of the circuitry, only that portion which is used tocut half of the screen master, either the upper or lower half, is

initially described. The additional logic circuits required to cut theentire screen surface are illustrated in FIG. 14.

I In FIG. 13, circuitry is disclosed for generating the electricalwaveform whereby the cutting stylus can be modulated to producemicroelements having depth profiles similar to those illustrated inFIGS. 5a and 5b. It has been found that the required electrical waveformcan be achieved by adding a sawtooth waveform, in varying amountsdepending upon the y-position of the cutting stylus, to the asymmetricalwaveform required for producing the desired depth profile at the screencenter (y 0). To generate the necessary sawtooth waveform, a sawtoothgenerator 130 is provided, such generator comprising a flip-flop 131,alimit detector 132, resistors R and R,, diode D1, capacitor C1 andoperational amplifier A1. Amplifier A1 is connected as an integrator togive a linear ramp while the voltage at input signal from the shapingcircuit only when V, is

negative (i.e., during the slow ramp). The output of the shaping circuitl40 produces a depth profile for the screen elements at'y 0. To producethe required depth profile as the y-position of the cutting stylusgradually increases, it is necessary to gradually tilt the y 0 depthprofile. Such gradual tilting is accomplished by gradually adding, asthe y-position of the cutting stylus gradually increases, the sawtoothwaveform V, to the output of the shaping circuit 104. Such addition isaccomplished by operational amplifier A2. To vary the contribution ofthe sawtooth waveform, resistor R3 is mechanically varied by they-position of the milling machine carriage. In FIG. 16, the output f ofthe amplifer A2 is illustrated when the cutting stylus is in a positiondisplaced along the y axis from screen center.

To provide an electrical waveform whereby the cutting stylus can bemodulated in such a manner as to properly vary the groove depth on bothsides of the screen center, it is necessary to provide circuitry forinterchanging the direction of the fast and slow slopes of the sawtooth(i.e., change the sense of the sawtooth) as the cutting stylus passesthrough the center of the screen. Moreover, it is necessary to switchthe shapeblanking so that gate 143 operates during the fastslopedportion of the sawtooth, whether positive or q, V,, is constant. When V,is negative, the ramp output V, of the sawtooth generator ispositive-going. When V, exceeds the positive threshold of limit detector132, flip-flop 131 is switched to make V, positive, at which time V,becomes negative-going. When V, reaches a negative limit, the flip-flopis again switched, to make V, negative again. Operational amplifier Alpreferably has complementary outputs, and the limit detector 132 acts bydetecting the negative limit of first one output and then the other. Theslopes of the positiveand negative-going ramps of the sawtooth arecontrolled by diode D1 and resistors R1 and R2, the latter beingvariable. Diode D1 is non-conducting when V, is negative, and conductsonly when V, is positive. Therefore, the positive-going ramp is slowerthan the negative-going ramp because the slope of the former isdetermined by the current flowing through resistor R1 only, whereas theslope of the latter is determined by the current flowing through bothresistors R1 and R2. By varying the value of resistor R2, the slopeofthe negative-going ramp can be varied.

The sawtooth output of generator 130 is then fed through a non-linearshaping circuit 140 which, as

negative-going. The required circuitry is illustrated in FIG. 14. To"switch thesense of the sawtooth at y O, the fast-sloped diode D1 ofFIG. 13 is replaced by an exclusive OR gate 150, the output of which iscontrolled by a flip-flop 151. The output of flip-flop 151 is controlledby a switch on the milling machine carriage, switching from one state toanother as the screen center (y 0) passes the cutting stylus. Toproperly cut both sides of the screen surface, it is necessary toprovide an additional shaping circuit 152 since the asymmetry caused bythe arcuate movement of the cutting stylus does not depend on theparticular portion of the screen being cut. The output of the propershaping circuit is supplied to amplifier A2, during the slow rampportion of the output of the sawtooth generator, through blanking gates143 and 153 which are controlled respectively by NAND gate 155 and ORgate 156.

To initiate the cutting operation, a start button is pressed whichpivots the cutting assembly 40 about pin 44 into a cutting position,causes the hydraulicpneumatic motor 95 to move the milling machinecarriage in the y direction and causes the above-described electroniccircuitry to drive the cutting stylus according to the waveform of theelectrical signal applied thereto. After cutting a groove ofpredetermined length, a microswitch (not shown) is actuated by carriage93 which serves to stop penumatic motor 95, activate a solenoid whichmoves cam 46 of the recording head clockwise into a position to pivotthe cutting assembly into a inoperative position, and actuate steppingmotor so as to move workbed 88 a predetermined distance in the xdirection. The microswitch also returns the milling machine carriage toits starting position on the y-axis which, in turn, actuates a secondmicroswitch. When actuated, the second microswitch rotates cam 46counterclockwise to permit the recording head to pivot into an operablecutting position, and the cutting process is repeated. This processcontinues without interruption until the entire screen master has beencut.

As the heated stylus S cuts a groove in the acetate coating, acontinuous silver or chip is extricated from the workpiece surface. Tocontinuously draw this silver away from the workpiece, a vacuum nozzle162 (shown in FIG. 7) connected to a vacuum source through hose 163, ispositioned adjacent stylus S during the cutting operation. The maximumdepth of cut produced by the stylus in the acetate coating is controlledby ball follower 51 which, as mentioned hereinabove, rides on an uncutportion of the coating, near the stylus. The recording head includes amechanism for maintaining the distance between the stylus tip and thebase of the ball constant. Preferably, the groove spacing and minimumgroove depth are set such that no land or flat areas exist betweenadjacent grooves.

Another projection screen embodying the present invention is illustratedin FIGS. 17-19 and is generally designated by the reference numeral 200.Like the aforedescribed screen, the surface of screen 200 comprises aplurality of contiguous microelements having boundaries defined in atransverse direction by cusp lines 201 formed by the intersecting edgesof adjacent linear grooves 202, and in a longitudinal direction byperiodic shifts occurring along lines 203 in the groove depth profilefrom concave toconvex. However, unlike the aforedescribed screen, thecontour of all microelements comprising the surface of scfeen 200 issubstantially identical, each being contoured such as to uniformlydistribute (i.e., distribute with constant luminance) normally incidentimage light throughout the same horizontal and vertical audience angles.Each mi-.

croelement has transverse and longitudinal cross sections defined by theabove equations. In order-to satisfy Postulate I set forth above, namelythat each elemental area on the screen surface distribute image lightonly throughout an angle just large enough to encompass a predefinedviewing area, it is necessary to orient each microelement on the screensurface such that the planes which bisects its vertical and horizontalaudience angles intersects with similar bisecting planes of all othermicroelements at a point remote from the screen surface. Preferably,such orientation is accomplished by giving the screen surface asubstantially spherical shape, the center of spherical curvaturecorresponding to the desired point of intersection of the optical axesof the microelements.

Rather than curving the screen surface to effectively adjust theorientation of each microelement, Postulate I could be satisfied byadjusting the angle at which image light impinges upon eachmicroelement. Such adjustment could be accomplished in the case of arear projection screen by providing the rear surface 205 of theprojection screen, that is, the surface closest to the projector, with aFresnel-like lens See FIGS. 18a and 19a. Obviously, combinations ofFresnel lenses and other screen curvatures, such as cylindrical, couldbe used.

To fabricate screens of the type illustrated in FIGS. 17-19, a flatmaster is initially made using the techniques and apparatus describedabove. From the master, a flexible negative matrix is made, which may besubsequently curved and used to generate correspondingly curved positiveprojection screens by the molding techniques described hereinbelow.

In fabricating the master, both channels of the abovedescribed stereosound recording head are fed electrical waveforms defined by Equations(1) and (2) above,

at 0 0. In FIG. 20, electronic circuitry is schematically illustratedfor generating the required waveform. As previously indicated, suchwaveform differs from a true sine wave in .that the peaks are flattened,relative to the lower-amplitude portions of the wave. Moreover, as alsomentioned hereinabove, to produce a groove depth profile in accordancewith a desired waveform, it is necessary to drive the cutting styluswith an asymmetrical waveform which is converted to the desired waveformby the arcuate stylus movement. To produce the waveform required, theoutput (sin x) of a conventional sinewave generator 209 is firstasymmetrically distorted by asymmetrical circuit 210 to compensate forthe arcuate stylus movement, and then shaped by shaping circuit 211 toproduce the waveform required for appropriately driving the cuttingstylus S. It has been found that by adding to the sine waveform a smallamount of its second harmonic, the requisite asymmetrical distortion canbe achieved. A squaring circuit 212, such as an analog multipliermodule, is used to generate the second harmonic waveform (sin 21:) fromthe fundamental. Capacitor C2 is used to eliminate the dc component ofthe squaring circuit output so as to produce a positiveandnegative-going signal. Since the midpoint of the resulting waveform lagsthe sin x waveform by it is necessary to feed the output of the sinewave generator through a simple RC phaselagging circuit 213. In thismanner, the two waves are added while in phase by operational amplifierA3. The amount of asymmetry in the output of amplifier A3 depends, ofcourse, on the peak-to-peak amplitude of the added second harmonic.Potentiometer P1 serves to vary the second harmonic amplitude prior tobeing added to the fundamental.

To produce the desired waveform from the asymmetrically distorted sinewave output of circuit 210, such output is fed to the shaping circuit211. The input sig nal to shaping circuit 211 is segmented by reason ofhaving to overcome successively the forward voltage drops across diodesD3-Dl2. Diodes D3-D7 and D8-D12 serve to segment the positiveandnegative going portions of the input signal, respectively. Operationalamplifier A4 serves to sum the contributions of the various segments toproduce a difference signal Ax having a waveform representing thedifference by which the desired waveform differs from the asymmetricallydistorted sine wave input. The contributions of the individual segmentsto the output of amplifier A4 are adjusted by varying the values ofresistors R5-R9. The output of amplifier A4 is adjustable in amplitudeby potentiometer P2. Resistors R10 and R11 and potentiometer P2 serve tocontrol the gain of amplifier A5. By simply adding the difference signalAx, which is of a polarity opposite that of the unshaped signal due tothe polarity reversing affect of amplifier A4, to the unshaped signal,the desired waveform for driving the cutting stylus is achieved. Suchaddition is performed by operational amplifier A5. Resistors R10 and R11and potentiometer P3 serve to control the gain provided by the summingamplifier A5. The output of amplifier A5 is then used as the inputs toboth driving coils of the stereo recording head to drive the cuttingstylus.

After making the projection screen master in accordance with theaforedescribed method and apparatus, projection screens can be producedtherefrom by making a negative matrix or master from the original, andcasting positive screens, in a resinous material, from the negativemaster. Preferably, the negative master is made from General ElectricRTV-60 silicone rubber which is prepared by adding 3 grams of dibutyltin dilaurate RTV curing catalyst to 2 pounds of the RTV-60 rubber,agitating the mixture with an electric stirrer for minutes and thenplacing it in a bell jar which is then' evacuated to a pressure of 150microns of mercury for about minutes. Upon fixing sidewalls to the edgeof the original master, the RTV rubber mixture can then be poured intothis mold so that no air is entrapped. After curing, the rubber mold canthen be used to cast projection screens.

To fabricate spherically or otherwise curved front projection screens ofthe type depicted in FIGS. 17-19, the rubber negative mold is disposedon an approximately curved support prior to casting. A maraglas resin,after being degassed, is then poured into the mold. After heating in anoven at 200 F for several hours to harden the resin, the casting can becoated with an aluminum coating to form a front projection screen.Refractive or rear projection screens can be made from a planar masterby incorporating a Fresnellike lens on the castings replicatedtherefrom.

As indicated above front and rear projection screens merely constitute apreferred embodiment of the invention and it should be understood thatother radiationredistributive devices, such as those already mentioned,as well as the many obvious variations and modifications of theinvention as described, are considered within the spirit and scope ofthe invention.

We claim:

1. A radiation-redistributive device having a surface defining aplurality of parallel grooves, each of said grooves having a depth whichundulates along the groove length to define a row of alternately concaveand convex optical microelements, each of said microelements beingcontoured to redistribute incident radiation in such a manner as toproduce substantially uniform radiance throughout predefined verticaland horizontal field angles and being disposed such that a plane whichbisects one of its field angles and which extends perpendicular to thegroove length intersects with similarly defined planes of all othermicroelements substantially along a first line which extends parallel tosaid surface and is in a plane perpendicular to the groove length.

2. The invention according to claim 1 wherein said surface iscylindrically curved about said first line.

3. The invention according to claim 1 wherein said grooves arerectilinear.

4. The invention according to claim 3 wherein said grooves arecontiguously arranged to form cusp lines at the juncture of adjacentgrooves, said cusp lines undulating in a plane perpendicular to saidsurface and extending substantially parallel to the groove length.

5. The invention according to claim 1 wherein each of said microelementsis disposed such that the plane which bisects the other of its fieldangles and which extends parallel to the groove length intersects withsimilarly defined planes of all microelements substantially along asecond line which perpendicularly intersects said first line and is in aplane parallel to the screen surface.

6. the invention according to claim 5 wherein said grooves arerectilinear.

7. The invention according to claim 5 wherein said surface iscylindrically curved about said second line.

8. The invention according to claim 5 wherein said surface issubstantially reflective so as to reflect radiation emanating from asource on one side of the screen to a field situated on the same side ofthe screen to a field situated on the same side of the screen as thesource.

9. The invention according to claim 5 wherein the screen, including saidsurface, is substantially transparent so as to refractradiationemanating from a source on one side of the screen to a field situated onthe opposite side of the screen from the source.

10. The invention according to claim 5 wherein said surface isspherically curved about the intersection of said first and secondlines.

11. A projection screen for presenting to a predefined field ofobservation an image projected thereon by a projecting apparatus, saidscreen having a surface defining a plurality of contiguous grooves, eachof said grooves having a depth which periodically undulates along thegroove length and thereby defines a row of alternately concave andconvex optical microelements, each of said grooves having a transversecross section which is substantially defined by at least a segment of afirst curve where w and u are the coordinates of said curve, w

being measured in a direction parallel to the path of incident imagelight, u being measured in a direction perpendicular to w and in theplane of the cross section; and n is the refractive index of themicroelement (n being 1 when the microelement is reflective); and w hasa value with the limits:

l s w 5 cos (B 0'), when g(w;n) is positive and the microelement isrefractive;

-l 5 w 2 cos (A 0'), when g(w;n) is negative and the microelement isrefractive;

-l 5 W5 cos (A 0), when g(w;n) is positive and the microelement isreflective; and

-1 w -cos (B 0), when g(w;n) is negative and the microelement isreflective;

where 0 is the projection in the u w plane of the angle formed by a lineextending parallel to incident image light, and the normal to the screensurface; n sin Olsin 0; and A and B are the audience angles, measuredfrom a normal to the screen surface and in the plane of the crosssection, through which incident image light must be distributed to justencompass said field of observation.

12. The invention according to claim 11 wherein each of saidmicroelements has a longitudinal cross section, taken in a planeparallel to the groove length and normal to said surface, substantiallydefined by at least a segment of said first curve, and A and Band u aremeasured in the plane of said longitudinal cross section, and by atleast a segment of a second curve where w has a value within thefollowing limits:

cos (A 0')s w 5 1, when flwm) is positive and the microelement isrefractive;

cos (B 0') s w 5 1, when flwm) is negative and the microelement isrefractive;

cos (B 0) s w s 1, when flwm) is positive and the microelement isreflective;

cos (A 0) 5 w 5 lwhenflwm) is negative and the microelement isreflective.

13. A rear projection screen for presenting an image projected thereonto a predefined audience space, said 'sc'r een comprising a sheet ofsubstantially transparent material having a first surface comprisingmeans defining a plurality of Contiguous grooves each of said grooveshaving a transverse cross section defined by at least a segment of thecurve where w and u are the curve coordinates, w being mea sured in adirection parallel to the path of incident image light, u being measuredin a direction perpendicular to w and in the plane of the cross section;n is the refractive index of said material, and w has a value within thelimits 1 l w s c'os (B when g(w;n) is positive, and

l g w cos (.4 0'), when g(w;n) is negative; where A and B are theangles, measured from the normal to said first surface in the plane ofthe cross section, through which each groove redistributes image lightto just encompass said audience space, and sin 0' sin 0/n, where 0 isthe projection in the u w plane of the angle formed by the lineextending parallel to incident image light and the normal to said firstsurface.

14. The invention according to claim 13 wherein said first surface issubstantially planar, and said sheet of material has a second surface,spaced from and extending substantially parallel to said first surface,comprising means defining a Fresnel-type lens.

15. The invention according to claim 14 wherein each of said grooves hasa depth which periodically undulates along the groove length to define arow of alternately concave and convex optical microelements.

16. A rear projection screen for presenting an image 16 projectedthereon to a predefined audience space, said screen comprising a sheetof substantially transparent material having a first surface comprisingmeans defining a plurality of contiguous grooves, each of said grooveshaving a transverse cross section defined by at least a segment of thecurve u if(w;n) 1-w WW2" cos vw where w and u are the curve coordinates,w being measured in a direction parallel to the path of incident imagelight and u being measured in a direction perpendicular to w in theplane of the cross section; n is the refractive index of said material,an w has a value within the following limits; I

cos (A 0") 5 w 5 I, when flwm is positive; and

cos(B -l 0'); w s l, whenflvvm) is negative; where A and B are theangles, measured from the nor mal to said first surface in the plane ofthe cross section, through which each groove redistributes' incidentimage light to just encompass said audience space, and sin 0' sin 0/n,where 0 is the projection in the u w plane of the angle formed by theline extending parallel to incident image light and the normal to saidfirst surface.-

17; The invention according to claim 16 wherein said first surface issubstantially planar, and said sheet of material has a second surface,spaced from and extending substantially parallel to said first surface,comprising means defining a Fresnel-type lens.

18. The invention according to claim 17 wherein each of said grooves hasa depth which periodically un dulates along the groove length to definea row of alternately concave and convex optical microelements.

' "UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No.3,75% C Dated Q August 28 1973 ie'n ofls) v James J DePalma and HowardF. Langworthy It is eertified that error apfieara in theabove-identified patent: and that said Letters Patent are herebycorrected as shown below:

Column 2; line 28, "raidation" Should read "radiation"; Colunm-2g line 52, designed" Should read desired- Column 3, line 20 "s ytlus" shouldread "stylus"; Column C ling 11M s hou-ld read .3 he+5v I Column o' line"angle" should read angles-1 Column 9, Q line 5, "intersecting" shouldread -interesting--; V Y

column 10, line 10 "low should rad "l lo- Column-ll, line???""sC'feen"should read --screen-;

Column l3, lines 15-16, "approximately should read appropriately- Columnl3, linev65, "theflfshould rea de- Th eg Column'l L, lines #5, "to afield situated on the same side of the screen I should be deleted;

I Columnql tyline 38,,that partof the formula reading: "(he)", shouldread "(A 6)"; a

Column l5 lines l6--l7 that part of the formula reading: 15w 3*" shouldread-'-' ''lw--. v

Signed and sealed thisjZSth day of 'Decembe'r 1973. J

(S Attest: I

EDWARD M.FLETCHER,JR. RENE D. TEGTMEYE R A ttestlng Officer I v I ActingCommissioner of Patents M nmTsnsTATss i AT ENT OFFICE J TIFICATE OFCORRECTION 15,794,813 7 Dated August 28. 1973 Patent No.

e efle) James J DePalma and Howard F. Langworthy It is certified thaterror appears in the above-identified patent and that said LettersPatent are hereby corrected as shown below:

Column 2, line 28, raidation" should read --radiation-; Column 2, line52, "designed" should read desired Column 3, line 20, sytlus should readstylus- Column 6, line 17, "'th' should read the -3 Column 6, line 18,angle should read "angles";

Column 9, line 5, "intersecting" should read -interesting--;

Column lO, line 1o, "lo l" should read mo- Column 11, line 27, "scfeen"should read --screen-;

Column 13, lines 15-16, "approximately should read -appropriately--5Column 13, line 65, "the" should read "The";

Column l L, lines P5, "to a field situated on the same side of thescreen should be deleted;

Column 14, line 38, thatpart of the formula reading:

" (A=6)" should read "(A e)";

Column 15, lines 16-17, that part of the formula reading: '-lw shouldread 'lw---.'

Signed and sealed this 25th day of December 1973. J

(SEAL) Attest:

EDWARD M.FLETCHER,JR. RENE D. TEGTMEYER Attestlng Officer ActingCommissioner of Patents

1. A radiation-redistributive device having a surface defining aplurality of parallel grooves, each of said grooves having a depth whichundulates along the groove length to define a row of alternately concaveand convex optical microelements, each of said microelements beingcontoured to redistribute incident radiation in such a manner as toproduce substantially uniform radiance throughout predefined verticaland horizontal field angles and being disposed such that a plane whichbisects one of its field angles and which extends perpendicular to thegroove length intersects with similarly defined planes of all othermicroelements substantially along a first line which extends parallel tosaid surface and is in a plane perpendicular to the groove length. 2.The invention according to claim 1 wherein said surface is cylindricallycurved about said first line.
 3. The invention according to claim 1wherein said grooves are rectilinear.
 4. The invention according toclaim 3 wherein said grooves are contiguously arranged to form cusplines at the juncture of adjacent grooves, said cusp lines undulating ina plane perpendicular to said surface and extending substantiallyparallel to the groove length.
 5. The invention according to claim 1wherein each of said microelements is disposed such that the plane whichbisects the other of its field angles and which extends parallel to thegroove length intersects with similarly defined planes of allmicroelements substantially along a second line which perpendicularlyintersects said first line and is in a plane parallel to the screensurface.
 6. the invention according to claim 5 wherein said grooves arerectilinear.
 7. The invention according to claim 5 wherein said surfaceis cylindrically curved about said second line.
 8. The inventionaccording to claim 5 wherein said surface is substantially reflective soas to reflect radiation emanating from a source on one side of thescreen to a field situated on the same side of the screen to a fieldsituated on the same side of the screen as the source.
 9. The inventionaccording to claim 5 wherein the screen, including said surface, issubstantially transparent so as to refract radiation emanating from asource on one side of the screen to a field situated on the oppositeside of the screen from the sourCe.
 10. The invention according to claim5 wherein said surface is spherically curved about the intersection ofsaid first and second lines.
 11. A projection screen for presenting to apredefined field of observation an image projected thereon by aprojecting apparatus, said screen having a surface defining a pluralityof contiguous grooves, each of said grooves having a depth whichperiodically undulates along the groove length and thereby defines a rowof alternately concave and convex optical microelements, each of saidgrooves having a transverse cross section which is substantially definedby at least a segment of a first curve u + or - g(w;n) + or - ( SquareRoot 1+w ( Square Root -w -2n) + cos 1 Square Root - w) where w and uare the coordinates of said curve, w being measured in a directionparallel to the path of incident image light, u being measured in adirection perpendicular to w and in the plane of the cross section; andn is the refractive index of the microelement (n being -1 when themicroelement is reflective); and w has a value with the limits: -1 < or= w < or = -cos2 (B + theta ''), when g(w;n) is positive and themicroelement is refractive; -1 < or = w < or = -cos2 (A - theta ''),when g(w;n) is negative and the microelement is refractive; -1 < or = w< or = -cos2 (A theta ), when g(w;n) is positive and the microelement isreflective; and -1 < or = w < or = -cos2 (B + theta ), when g(w;n) isnegative and the microelement is reflective; where theta is theprojection in the u - w plane of the angle formed by a line extendingparallel to incident image light, and the normal to the screen surface;n sin theta /sin theta ''; and A and B are the audience angles, measuredfrom a normal to the screen surface and in the plane of the crosssection, through which incident image light must be distributed to justencompass said field of observation.
 12. The invention according toclaim 11 wherein each of said microelements has a longitudinal crosssection, taken in a plane parallel to the groove length and normal tosaid surface, substantially defined by at least a segment of said firstcurve, and A and B and u are measured in the plane of said longitudinalcross section, and by at least a segment of a second curve u + or -f(w;n) + or - ( Square Root 1-w ( Square Root w + 2n) + cos 1 SquareRoot w) where w has a value within the following limits: cos2 (A - theta'') < or = w < or = 1, when f(w;n) is positive and the microelement isrefractive; cos2 (B + theta '') < or = w < or = 1, when f(w;n) isnegative and the microelement is refractive; cos2 (B + theta ) < or = w< or = 1, when f(w;n) is positive and the microelement is reflective;cos2 (A - theta ) < or = w < or = 1when f(w;n) is negative and themicroelement is reflective.
 13. A rear projection screen for presentingan image projected thereon to a predefined audience space, said screencomprising a sheet of substantially transparent material having a firstsurface comprising means defining a plurality of contiguous grooves eachof said grooves having a transverse cross section defined by at least asegment of the curve u + or - g(w;n) + or - ( Square Root 1+w ( SquareRoot -w -2n) + cos 1 Square Root - w) where w and u are the curvecoordinates, w being measured in a direction parallel to the path ofincident image light, u being measured in a direction perpendicular to wand in the plane of the cross section; n is the refractive index of saidmaterial, and w has a value within the limits -1 < or = w < or =-cos2(B + theta ''), when g(w;n) is positive, and -1 < or = w < or =-cos2(A - theta ''), when g(w;n) is negative; where A and B are theangles, measured from the normal to said first surface in the plane ofthe cross section, through which each groove redistributes image lightto just encompass said audience space, and sin theta '' sin theta /n,where theta is the projection in the u - w plane of the angle formed bythe line extending parallel to incident image light and the normal tosaid first surface.
 14. The invention according to claim 13 wherein saidfirst surface is substantially planar, and said sheet of material has asecond surface, spaced from and extending substantially parallel to saidfirst surface, comprising means defining a Fresnel-type lens.
 15. Theinvention according to claim 14 wherein each of said grooves has a depthwhich periodically undulates along the groove length to define a row ofalternately concave and convex optical microelements.
 16. A rearprojection screen for presenting an image projected thereon to apredefined audience space, said screen comprising a sheet ofsubstantially transparent material having a first surface comprisingmeans defining a plurality of contiguous grooves, each of said grooveshaving a transverse cross section defined by at least a segment of thecurve u + or - f(w;n) + or - ( Square Root 1-w ( Square Root w +2n) +cos 1 Square Root w) where w and u are the curve coordinates, w beingmeasured in a direction parallel to the path of incident image light andu being measured in a direction perpendicular to w in the plane of thecross section; n is the refractive index of said material, an w has avalue within the following limits; cos2(A - theta '') < or = w < or = 1,when f(w;n) is positive; and cos2(B + theta '') < or = w < or = 1, whenf(w;n) is negative; where A and B are the angles, measured from thenormal to said first surface in the plane of the cross section, throughwhich each groove redistributes incident image light to just encompasssaid audience space, and sin theta '' sin theta /n, where theta is theprojection in the u - w plane of the angle formed by the line extendingparallel to incident image light and the normal to said first surface.17. The invention according to claim 16 wherein said first surface issubstantially planar, and said sheet of material has a second surface,spaced from and extending substantially parallel to said first surface,comprising means defining a Fresnel-type lens.
 18. The inventionaccording to claim 17 wherein each of said grooves has a depth whichperiodically undulates along the groove length to define a row ofalternately concave and convex optical microelements.